## Resonances

When an oscillating system is subject to an excitation (a periodic perturbation), it may experience an increase in the energy of oscillation. This phenomenon, known as resonance, depends on the excitation frequencies and the natural frequencies of the system.

We speak of orbital resonances when the phenomenon is the motion of a planet around a star (or a satellite around a planet). The excitation is the gravitational perturbation by another planet (or by another satellite).

Several parameters in the elliptical orbit may give rise to resonances:

• The primary one is the motion of the planet along its orbit. The mean velocity is n (known as the 'mean motion'). A mean-motion resonance occurs if the mean motions of two planets n1 and n2 form a rational ratio, i.e., if:

This configuration is frequently found in the Solar System:

Such a relationship may involve several objects. The satellites of Jupiter, Io, Europa, and Ganymede are linked by the relationship:

Resonances may give rise to a paradoxical effect: despite the fact that gravitation is a force of attraction, the cumulative effect of resonance may lead to a force of repulsion. When a body orbits within a ring or disk, the particles of the ring are subject to resonances of higher and higher order as they approach the body. The combined effect of these resonances is a force of repulsion, which causes the edge of the ring to recede from the body's orbit.

• Resonances may also link the mean motion of a body and its rotation. A spectacular example is the synchronous rotation of many of the planetary satellites and, in particular, of the Moon. The mean motion of the Moon around the Earth is equal to its rotation velocity, which causes it to present the same face to the Earth at all times. Tidal forces have created frictional forces within the Moon, which have braked its rotation until it was in synchronous rotation, a configuration that causes the friction to disappear.

• Resonances may also link the motion of periapses or ascending nodes. These resonances are said to be 'secular' because these motions are slower than the mean motion.